# H"olderian error bounds and Kurdyka-Lojasiewicz inequality for the trust region subproblem

arXiv: https://arxiv.org/abs/1911.11955 · HTML: https://mlsanity.com/paper/1911.11955/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Nov 2019.

## Built on by (most cited first, 2 of 2)

- [On the Complexity of Finite-Sum Smooth Optimization under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/2402.02569/) (2024)
- [Error Bound Analysis for the Regularized Loss of Deep Linear Neural Networks](https://mlsanity.com/paper/2502.11152/) (2025)

## Abstract (arXiv)

In this paper, we study the local variational geometry of the optimal solution set of the trust region subproblem (TRS), which minimizes a general, possibly nonconvex, quadratic function over the unit ball. Specifically, we demonstrate that a H"olderian error bound holds globally for the TRS with modulus 1/4 and the Kurdyka-Lojasiewicz (KL) inequality holds locally for the TRS with a KL exponent 3/4 at any optimal solution. We further prove that unless in a special case, the H"olderian error bound modulus, as well as the KL exponent, is 1/2. Finally, based on the obtained KL property, we further show that the projected gradient methods studied in [A. Beck and Y. Vaisbourd, SIAM J. Optim., 28 (2018), pp. 1951--1967] for solving the TRS achieve a sublinear or even linear rate of convergence.